Final answer:
To compute sample variance and standard deviation, one must first calculate the sample mean, then use the sum of squared deviations divided by the number of data points minus one. Both the standard and alternative methods yield a sample variance of 10.8 and a standard deviation of approximately 3.3.
Step-by-step explanation:
Finding the Sample Variance and Standard Deviation
To find the sample variance and standard deviation using the sample variance definition, you first need to calculate the sample mean.
Add all the values together: 5 + 9 + 8 + 5 + 12 + 3 = 42.
Divide by the number of data points to get the mean: 42 / 6 = 7.
Subtract the mean from each value and square the result: (5-7)^2 + (9-7)^2 + (8-7)^2 + (5-7)^2 + (12-7)^2 + (3-7)^2 = 4 + 4 + 1 + 4 + 25 + 16 = 54.
Divide this sum by the number of data points minus one (n-1) for the sample variance: 54 / (6-1) = 54 / 5 = 10.8.
The standard deviation is the square root of the variance, so √10.8 ≈ 3.3 (rounded to one decimal place).
Using the alternative formula for sample variance involves first finding the square of all values, summing them up, and then subtracting the square of the mean times the number of observations, all divided by (n-1).
Find the square of all values and sum: 5^2 + 9^2 + 8^2 + 5^2 + 12^2 + 3^2 = 25 + 81 + 64 + 25 + 144 + 9 = 348.
Subtract the square of the mean multiplied by the number of data points: 348 - (7^2 * 6) = 348 - 294 = 54.
Divide by (n-1) to find the variance: 54 / 5 = 10.8.
Find the standard deviation by taking the square root of the variance: √10.8 ≈ 3.3.
Both methods yield the same result for variance (10.8) and standard deviation (3.3).